A Morita map is a groupoid morphism \(\phi :(G\rightrightarrows M) \rightarrow (G^{\prime }\rightrightarrows M^{\prime })\)  which is fully faithful and essentially surjective, in the sense that the source/target maps define a fibred product of manifolds \(G \cong (M \times M) \times _{ (M^{\prime }\times M^{\prime })} G^{\prime }\)  and that the map \(G^{\prime } \times _{M^{\prime }}M \rightarrow M\)  sending \((\phi ^0(x)\rightarrow y)\mapsto y\)  is a surjective submersion, see [1]}, [2]}. An important fact shown in [1]} is that a Lie groupoid morphism is a Morita map if and only if it yields an isomorphism between transversal data. That is, the morphism must induce: a homeomorphism between the orbit spaces, a Lie group isomorphism \(G_x\cong G^{\prime }_{\phi ^0(x)}\)  between the isotropies and isomorphisms between the normal representations \(G_x\curvearrowright \nu _x\rightarrow G^{\prime }_{\phi ^0(x)}\curvearrowright \nu ^{\prime }_{\phi ^0(x)}\) .
